252 Variations of the Arbitrary Constants in Elliptic Motion. 



The squares of (7) when added, by putting c'' +c'^+&'-—b% 

 ldx^^-\-dy^^dz^\ Irdry dR 



giver^( ii j-Vdl] =^"' (1^)' "^^° P"^ ^rf^ + 



y -r- -j-2-j-=rR', muhiply (5) by a;, y, sr, add the products, and 



xd^x4-vd^y-\-zd''z M ^ / ^x , ■, 



we get —dF^ +y+rR'=0, (19), by xdx+ydy + 



xd~x-{-yd^y-\-zd^z d(rdr) 

 zdz = rdr, and (18), we have —^^ ~ ~dF~ " 



dx^+dy^+dz^ rd'r b^ /,^x x. ^''' , 



— -j^l =-^^-^' (20); hence (19) becomes ^ + 



Mr-62 dR" d'b'- 



^:^+R'=0, (21),putMr-&-^=--.,-^=MR'+^^' (22), 



dH Ms dR" 

 and (21) will be changed to ^ + 77+"^=0, (23). 



^R ^_^ ^R dR^_df dR dR'' df" 



dx ds dt dy ^ ds dt ' dz '^ ds dt 



(24), then multiply (23) by x, and the first of (5) by —s, add the 



xd^s — sd^x 

 products, and we get after multiplying by dt, -r. = df, whose 



xds — sdx yds — sdy zds — sdz 



integral is — ^— =/, similarly —7^— =f, — ^r^ =/". 



(25), these give cs=yf-xf, c's=zf-xf", c"s=zf'—yf", (26), 

 by the first two of these s(cy' — c/''') =2//'' — J///", which compa- 

 red with the third gives cf"-{-fc" — c'f'=^0, (27) ; we also have by 

 (25) cds=fdy-f'dx, c'ds=fdz-f"dx, c"ds=f'dz-f"dy, (28), 

 then by taking the differentials of (26) having regard to (28), we 

 getsdc=ydf-xdf, sdc'=zdf-xdf", sdc" = zdf - ydf" , (29). 



Uf - cj"\ 

 By the first and second of (26) we havec^: — c'y4- 1 f — )a;=0, 



which reduces by (27) to cz- c'y-\-c"x=Q, and agrees with (8), as 

 it evidently ought to do. 



Now, since r^ =^x'^ ■\-y'^ -^-z"-, by restoring the value of 5 in (26), 

 then substituting the value of z from (8) in the first, the value of 

 y in the second, and the value of x in the third, we shall have three 

 equations of the second order; the first in terms oi x and y, the sec- 

 ond in terms of x and z, the third in terms of y and z ; which show 

 that m is constantly moving in some conic section, whose elements 

 are constantly changing, since c, c', c", f, f, f" are continually va- 



